Existence theorems for a nonlinear second-order distributional differential equation

In this work, we are concerned with existence of solutions for a nonlinear second-order distributional differential equation, which contains measure differential equations and stochastic differential equations as special cases. The proof is based on the Leray--Schauder nonlinear alternative and Kurzweil--Henstock--Stieltjes integrals. Meanwhile, examples are worked out to demonstrate that the main results are sharp.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of King Saud University - Science', ISSN 1018-3647. Submitted 05-March-2017; revised 24-April-2017; accepted for publication 26-April-201

Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

This paper is concerned with the existence of solutions for a kind of nonhomogeneous critical p-Kirchhoff type problem driven by an integro-differential operator L p K . In particular, we investigate the equation: M �ZZ R2n |v(x) − v(y)| p |x − y| n+ps dxdy� L p K v(x) = µg(x)|v| q−2 v + |v| p s −2 v + µ f(x) in R n where g(x) > 0, and f(x) may change sign, µ > 0 is a real parameter, 0 ps, 1 < q < p < p s , p s = np n−ps is the critical exponent of the fractional Sobolev space W s,p K (Rn ). By exploiting Ekeland’s variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact that M may be zero and lack of compactness at critical level L p s (Rn Our conclusions improve the related results on this topic

Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

This paper is concerned with the existence of solutions for a kind of nonhomogeneous critical $p$-Kirchhoff type problem driven by an integro-differential operator $\mathcal{L}^{p}_{K}$. In particular, we investigate the equation: \begin{align*} \mathcal{M}\left(\iint_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+ps}}dxdy\right) \mathcal{L}^{p}_{K}v(x)=\mu g(x)|v|^{q-2}v+|v|^{p_{s}^{*}-2}v+\mu f(x) \quad\mbox{in}~\mathbb{R}^{n}, \end{align*} where $g(x)>0$, and $f(x)$ may change sign, $\mu>0$ is a real parameter, $0ps$, $1<q<p<p_{s}^{*}$, $p_{s}^{*}=\frac{np}{n-ps}$ is the critical exponent of the fractional Sobolev space $W^{s,p}_{K}(\mathbb{R}^{n}).$ By exploiting Ekeland's variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact that $\mathcal{M}$ may be zero and lack of compactness at critical level $L^{p_{s}^{*}}(\mathbb{R}^{n})$. Our conclusions improve the related results on this topic

Some characterizations of the primitive of strong Henstock-Kurzweil integrable functions

summary:In this paper we give some complete characterizations of the primitive of strongly Henstock-Kurzweil integrable functions which are defined on $\mathbb{R}^m$ with values in a Banach space

Existence of Solutions of Second Order Boundary Value Problems with Integral Boundary Conditions and Singularities

By the notation and monotone convergence theorem of Henstock-Kurzweil integral, we investigate the existence of continuous solutions for the second order boundary value problems with integral boundary conditions in which the nonlinearities are allowed to have the singularities in t and are not Lebesgue integrable.</p

Equations containing locally Henstock-Kurzweil integrable functions

summary:A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions

© Hindawi Publishing Corp. ON HENSTOCK-DUNFORD AND HENSTOCK-PETTIS INTEGRALS

Abstract. We give the Riemann-type extensions of Dunford integral and Pettis integral, Henstock-Dunford integral and Henstock-Pettis integral. We discuss the relationships be-tween the Henstock-Dunford integral and Dunford integral, Henstock-Pettis integral and Pettis integral. We prove the Harnack extension theorems and the convergence theorems for Henstock-Dunford and Henstock-Pettis integrals

On Henstock-Dunford and Henstock-Pettis integrals

We give the Riemann-type extensions of Dunford integral and Pettis integral, Henstock-Dunford integral and Henstock-Pettis integral. We discuss the relationships between the Henstock-Dunford integral and Dunford integral, Henstock-Pettis integral and Pettis integral. We prove the Harnack extension theorems and the convergence theorems for Henstock-Dunford and Henstock-Pettis integrals

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p·-fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results